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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"4799271d3936beb013372510e6ce44bfa5568b7b288ff2b111b6ec5260cc687b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0135f5d413e6cc58","role":"sequence","name":"basis_map_A007318 of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     7,      3;\n  2:    26,     25,      5;\n  3:    85,    126,     56,      7;\n  4:   265,    531,    369,    102,      9;\n  5:   804,   2035,   1936,    847,    165,     11;\n  6:  2393,   7332,   8892,   5382,   1677,    247,    13;\n  7:  7019,  25275,  37350,  29040,  12630,   3003,   350,   15;\n  8: 20350,  84269, 147067, 140114,  78795,  26350,  4998,  476,  17;\n  9: 58445, 273714, 551380, 623029, 431433, 187625, 50388, 7866, 627, 19;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3+4*n^2-14*n+5+(16*n^2-10*n+1)*x+(8*n-4)*x^2)*P(n,x) = (32*n^3+40*n^2-74*n+23+(24*n^3+92*n^2-64*n+6)*x+(48*n^2+26*n-25)*x^2+(24*n-12)*x^3)*P(n-1,x) - (20*n^2+50*n+5+(36*n^2+130*n+44)*x+(16*n^2+112*n+87)*x^2+(32*n+64)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^3 + 40*n^2 - 74*n + 23 + (24*n^3 + 92*n^2 - 64*n + 6)*x + (48*n^2 + 26*n - 25)*x^2 + (24*n - 12)*x^3)*P[n - 1, x] - (20*n^2 + 50*n + 5 + (36*n^2 + 130*n + 44)*x + (16*n^2 + 112*n + 87)*x^2 + (32*n + 64)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 + 4*n^2 - 14*n + 5 + (16*n^2 - 10*n + 1)*x + (8*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((32*n^3 + 40*n^2 - 74*n + 23 + (24*n^3 + 92*n^2 - 64*n + 6)*x + (48*n^2 + 26*n - 25)*x^2 + (24*n - 12)*x^3)*v[i - 1] - (20*n^2 + 50*n + 5 + (36*n^2 + 130*n + 44)*x + (16*n^2 + 112*n + 87)*x^2 + (32*n + 64)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 + 4*n^2 - 14*n + 5 + (16*n^2 - 10*n + 1)*x + (8*n - 4)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9496336e2b29e28e","steps":["x \u21a6 x+1","D+1"],"edges":["e-557cf165357d4e1f","e-5180f8c68ec17ed2"],"text":"A082985 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A082985 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}